Chapter 29

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove each of the following : Any element algebraic over \(K\) is algebraic over \(F\), and conversely.

Let \(F\) be a field of characteristic \(\neq 2\). Let \(a \neq b\) be in \(F\). Prove the following: Any field \(F\) containing \(\sqrt{a}+\sqrt{b}\) also contains \(\sqrt{a}\) and \(\sqrt{b}\). [HINT: Compute \((\sqrt{a}+\sqrt{b})^{2}\) and show that \(\sqrt{a b} \in F\). Then compute \(\sqrt{a b}(\sqrt{a}+\sqrt{b})\), which is also in \(F .]\) Conclude that \(F(\sqrt{a}+\sqrt{b})=F(\sqrt{a}, \sqrt{b})\).

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove each of the following : If \(b\) is algebraic over \(K\), then \([F(b): F] \mid[K(b): F]\).

Let \(F\) be a field of characteristic \(\neq 2\). Let \(a \neq b\) be in \(F\). Prove the following: If \(b \neq x^{2} a\) for any \(x \in F\), then \(\sqrt{b} \notin F(\sqrt{a}) .\) Conclude that \(F(\sqrt{a}, \sqrt{b})\) is of degree 4 over \(F\).

Show that every element of \(\mathbb{R}(2+3 i)\) can be written as \(a+b i\), where \(a, b \in \mathbb{R}\). Conclude that \(\mathbb{R}(2+3 i)=\mathbb{C}\).

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove each of the following : If \(b\) is algebraic over \(K\), then \([K(b): K] \mid[F(b): F]\). (HINT: The minimum polynomial of \(b\) over \(F\) may factor in \(K[x]\), and \(b\) will then be a root of one of its irreducible factors.)

Let \(F\) be a field of characteristic \(\neq 2\). Let \(a \neq b\) be in \(F\). Prove the following: Show that \(x=\sqrt{a}+\sqrt{b}\) satisfies \(x^{4}-2(a+b) x^{2}+(a-b)^{2}=0\). Show that \(x=\sqrt{a+b+2 \sqrt{a b}}\) also satisfies this equation. Conclude that $$ F(\sqrt{a+b+2 \sqrt{a b}})=F(\sqrt{a}, \sqrt{b}) $$

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove each of the following : If \(b\) is algebraic over \(K\), then \([K(b): F(b)] \mid[K: F\rceil\).

Find a basis of \(\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})\) over \(\mathbb{Q}\), and describe the elements of \(\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})\).

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove each of the following : Let \(p(x)\) be irreducible in \(F[x]\). If \([K: F]\) and deg \(p(x)\) are relatively prime, then \(p(x)\) is irreducible in \(K[x]\)

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove the following : If an irreducible polynomial \(p(x) \in F[x]\) has a root in \(K\), then \(\operatorname{deg} p(x) \mid[K: F]\).

Find a basis of \(\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})\) over \(\mathbb{Q}\), and describe the elements of \(\mathbb{Q}(\sqrt{2}, \sqrt{3},\), \(\sqrt{5}\).